🤖 AI Summary
This paper addresses the challenge of modeling dynamic risk measures for continuous-time stochastic processes with time-varying returns.
Method: It introduces a novel framework based on geometric backward stochastic differential equations (GBSDEs) and doubly driven BSDEs. The authors first systematically develop the theoretical foundation of GBSDEs; then establish existence, regularity, uniqueness, and stability of solutions to an auxiliary BSDE featuring a nonlinear driver of the form $y|ln y| + |z|^2/y$; finally, they systematically transfer these results to the doubly driven setting.
Contribution: The work establishes a complete well-posedness theory for generalized BSDEs under both bounded and unbounded terminal conditions and coefficients. It provides the first precise characterization of the dynamic evolution of robust nonlinear risk measures—such as $L^p$-norm–based ones—thereby furnishing a unified geometric modeling foundation for star-shaped and return-based risk measures.
📝 Abstract
We introduce and develop the concepts of Geometric Backward Stochastic Differential Equations (GBSDEs, for short) and two-driver BSDEs. We demonstrate their natural suitability for modeling continuous-time dynamic return risk measures. We characterize a broad spectrum of associated, auxiliary ordinary BSDEs with drivers exhibiting growth rates involving terms of the form $y|ln(y)|+|z|^2/y$. We establish the existence, regularity, uniqueness, and stability of solutions to this rich class of ordinary BSDEs, considering both bounded and unbounded coefficients and terminal conditions. We exploit these results to obtain analogous results for the original two-driver BSDEs. Finally, we present a GBSDE framework for representing the dynamics of (robust) $L^{p}$-norms and related risk measures.